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:This article needs to be clarified and expanded to more clearly describe and discuss the several different concepts of "angle" which are in common use. This is something I plan to do eventually; I've been occasionally gathering sources at [[User:Jacobolus/Angle]] but am not yet ready to write a solid survey. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 00:45, 10 November 2023 (UTC)
Thank you for those sources on directed angle. The issue of angle definition came to a head in 1893 in Chicago when Felix Klein shot down Alexander Macfarlane’s paper on the topic which was presented to the mathematical congress held in connection with the [[Columbia Exposition]]. The paper was included in ''Papers in Space Analysis'' (1894) as the ''Proceedings'' of the congress only noted the title. The notion of an area-based definition referring to the sector of a circle was included in ''The Elements of Plane Trigonometry'' (1892) by R. Levitt & C. Davison (page 158). [[Robert Baldwin Hayward]] noted the complete analogy between the circular and hyperbolic cases when area of sectors are used to define the circular and hyperbolic angles. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 01:19, 12 November 2023 (UTC)
:Of course, both circular and hyperbolic angle measure can be defined either via area or via arc length, with the latter using a Lorentzian (pseudo-Euclidean) concept of distance in the plane. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:25, 12 November 2023 (UTC)
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